If you look at how cults or religions got started in recent history, we never see a bunch of people coordinate in that kind of a way, where a lot of people disparately decide to invent a nonexistent leader and go around telling everyone that they recently died.
Could the central authority have later fabricated a history of the early church that made it look like there was a long period where nobody could have coordinated big changes? Maybe, but that is a standard that no set of historical documents could pass. (It's like suggesting that the Byzantines fabricated every single classical document including the tree-rings of ancient historians arguing with each other.)